Digital filtering of discrete-time sampled signals is used extensively in commercial applications such as compact disc players, digital stereo systems, digital mixing boards, digital speaker crossover networks, and in many other audio and non-audio signal processing applications. In order to make digital filters practical for real-time applications, they must have sufficiently low computational complexity. Additionally, many applications require a linear-phase response in order to avoid frequency dependent temporal distortions.
Present technology employs two major types of digital signal filters: infinite impulse response (IIR) filters and finite impulse response (FIR) filters (see Oppenheim & Schafer, Discrete-Time Signal Processing, Prentice Hall, 1989, which is incorporated herein by reference). IIR filters are fast, due to their computational simplicity, but introduce temporal distortions into signals because they do not have linear-phase response. FIR filters, on the other hand, can be designed to have linear-phase response and hence no temporal distortion, but are generally slow due to the large number of arithmetic operations performed per sample of input. Clearly, it would be desirable to design a filter that combines the computational efficiency of an IIR filter with the linear-phase response of an FIR filter.
Oppenheim & Schafer, Digital Signal Processing, Prentice Hall, 1975, p. 161, describe an FIR filter which has the computational efficiency of an IIR filter. To avoid the stability difficulties inherent in their approach, however, they introduce a scaling factor which destroys the linear-phase response of the filter. Moreover, their approach does not allow for the implementation of filters with repeated poles.
One possibility for obtaining such a combination is to improve the computational efficiency of an FIR filter by taking advantage of the computationally efficient features of IIR filters while also retaining the desirable linear-phase properties. This combination of features may be accomplished by truncating the response of an IIR filter to obtain a truncated infinite impulse response (TIIR) filter.
This approach was used by Fam and Saramaki, Proc. IEEE Int. Symposium on Circuits and Systems, 1990, p. 3271, to obtain a limited class of linear-phase response FIR filters having the efficiency of IIR filters. Their method, however, has several disadvantages. The class of filters that is constructable by their methods is restricted to filters with pure exponential impulse responses. In particular, their method can not be used to construct truncated polynomial-times-exponential response filters, which have several important applications, including high-resolution frequency estimation. Moreover, their method of constructing and implementing filters is complicated and, consequently, is not amenable to generating or adjusting filter parameters in real-time applications. Nor do they suggest ways to make efficient FIR filters time-varying.